Factor Label Review
For any equation, as long as you know the units, “factor labeling” can be used to solve the problem. Remember, this is basically like a puzzle. You are going from one unit to another until you get to the unit that is wanted. Conversion factors and ratios are what allow you to go from one unit to another. Again, these are numeric relationships between different units.
Sample Problem 1
How many meters are in 3.5 kilometers?
1 1 km
(Remember from math, if the same number or unit is over itself this equals 1; which is why the given unit is cancelled)
Sample Problem 2
Aluminum has a density of 2.70 g/cm³. If there is a block of aluminum that occupies 4.5cm3, how many grams is it?
1 1 cm³
Sample Problem 1
How many meters are in 3.5 kilometers?
- You have to go from kilometers to meters using a conversion factor. There is only one conversion factor: 1 km = 1000 m.
- Factor labeling (aka dimensional analysis) requires the canceling of units (i.e., cross canceling). This is done by orienting the conversion factor such that the given unit is on the bottom …
1 1 km
(Remember from math, if the same number or unit is over itself this equals 1; which is why the given unit is cancelled)
Sample Problem 2
Aluminum has a density of 2.70 g/cm³. If there is a block of aluminum that occupies 4.5cm3, how many grams is it?
- On this problem, you have to go from cm³ (volume) to grams (mass) using a conversion factor. There is only one conversion factor that involves mass and volume: density.
- Just like before, cancel your units by orienting the conversion factor such that the given unit is on the bottom …
1 1 cm³
In some problems, more than one step may be required. Notice, though, that in each step, the unit given is placed on the bottom and the unit wanted winds up on top.
Sample Problem 3
33 mm = ___ hm
1 1 mm 100 m
**Notice here, that we are converting from one prefix to another prefix; so the middle step is converting to the base unit. The base unit is what both prefixes have in common.
Sample Problem 4
How many gallons are in 225 milliliters?
1 1 mL 3.785 L
One last point for converting squared or cubed units. The factor label steps do not change. You still place the unit given on the bottom and the unit wanted on top. What is affected, however, are the numbers put in front of each unit.
Sample Problem 5
3 cm³ = ___ in³
1 2.54 cm
**You should notice something: “cm” do not cancel “cm³ ”. So, you have to cube the unit to cancel it. You must cube the number to cube the unit.
3 cm³ x (1 in)³ = This is always done, even if several steps are required.
1 (2.54 cm)³
Sample Problem 3
33 mm = ___ hm
- You have to go from millimeters to meters, then from meters to hectometers (because each metric prefix has a numeric value; it means a number)
- Just as you have done, cross cancel the units by placing your wanted unit on top and your given unit on the bottom.
1 1 mm 100 m
**Notice here, that we are converting from one prefix to another prefix; so the middle step is converting to the base unit. The base unit is what both prefixes have in common.
Sample Problem 4
How many gallons are in 225 milliliters?
- You have to identify the unit that gallons and milliliters have in common, i.e., liters. So our conversion factors are : 1mL = 0.0001 L ; 3.785 L = 1 gal
- Once again, our plan has not changed – the given unit is placed on the bottom and the wanted unit is placed on top in all steps.
1 1 mL 3.785 L
One last point for converting squared or cubed units. The factor label steps do not change. You still place the unit given on the bottom and the unit wanted on top. What is affected, however, are the numbers put in front of each unit.
Sample Problem 5
3 cm³ = ___ in³
- You have to go from centimeters to inches
- Just as you have done, cross cancel the units by placing your wanted unit on top and your given unit on the bottom.
1 2.54 cm
**You should notice something: “cm” do not cancel “cm³ ”. So, you have to cube the unit to cancel it. You must cube the number to cube the unit.
3 cm³ x (1 in)³ = This is always done, even if several steps are required.
1 (2.54 cm)³